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Conjugacy invariants of Möbius transformations. (English) Zbl 0721.30035

This paper is devoted to generalizing the trace-squared (conjugation) invariant of Möbius transformations with complex entries to elements of a certain set of Clifford matrices ${𝒞}^{n}$. These were first considered by K. Th. Vahlen [Math. Ann. 55, 585-593 (1902)] and later by L. V. Ahlfors [see e.g. Ann. Acad. Sci. Fenn., Ser. A I 105, 15-27 (1985; Zbl 0586.30045)]. The Clifford matrices considered here are slightly different from those obtained by Ahlfors and provide a generalization to orientation-reversing maps. The main theorem is that, for each $k=0,1,2,···,n+2$, the ${T}_{k}$ function (the definition is too complicated to give here)

${T}_{k}:M\left(2,{𝒞}^{n}\right)\to ℝ$

is a conjugation invariant. Here ${𝒞}^{n}$ is the algebra of Clifford numbers generated by n roots of -1. The conjugation is by Clifford matrices. Among the properties of the invariant ${T}_{k}$ are the following:

(i) ${T}_{k}\left(-g\right)=T\left(g\right)$ for all $g\in {𝒞}^{n}$. Thus ${T}_{k}$ is well-defined on the projectivization of ${𝒞}^{n}·$

(ii) ${T}_{k}\left(g\right)=0$ for all k odd (resp. even) if g is orientation preserving (resp. reversing)

$\sum _{0}^{n+2}{T}_{k}\left(g\right)=0·$

(iii) $g\in {𝒞}^{n}$ is hyperbolic if and only if ${T}_{k}\left(g\right)<0$ for the largest k so that ${T}_{k}\left(g\right)\ne 0·$

(iv) $g\in {𝒞}^{n}$ is elliptic if and only if the quadratic form $N\left(z\right):=z\overline{z}$, when restricted to the kernel of Id-r(g) is not positive semidefinite. Here r takes g from the isometries of the ball model of hyperbolic space to the hyperboloid model.

##### MSC:
 30G35 Functions of hypercomplex variables and generalized variables